Tensor train based sampling algorithms for approximating regularized Wasserstein proximal operators

We present a tensor train (TT) based algorithm designed for sampling from a target distribution and employ TT approximation to capture the high-dimensional probability density evolution of overdamped Langevin dynamics. This involves utilizing the regularized Wasserstein proximal operator, which exhibits a simple kernel integration formulation, i.e., the softmax formula of the traditional proximal operator. The integration, performed in $\mathbb{R}^d$, poses a challenge in practical scenarios, making the algorithm practically implementable only with the aid of TT approximation. In the specific context of Gaussian distributions, we rigorously establish the unbiasedness and linear convergence of our sampling algorithm towards the target distribution. To assess the effectiveness of our proposed methods, we apply them to various scenarios, including Gaussian families, Gaussian mixtures, bimodal distributions, and Bayesian inverse problems in numerical examples. The sampling algorithm exhibits superior accuracy and faster convergence when compared to classical Langevin dynamics-type sampling algorithms.